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On Lie Algebras of Generalized Jacobi Matrices

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On Lie Algebras of Generalized Jacobi Matrices **************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2020 ON LIE ALGEBRAS OF GENERALIZED JACOBI MATRICES ALICE FIALOWSKI University of P´ecsand E¨otv¨osLor´andUniversity Budapest, Hungary E-mail: fi[email protected], fi[email protected] KENJI IOHARA Univ Lyon, Universit´eClaude Bernard Lyon 1 CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France E-mail: [email protected] Abstract. In these lecture notes, we consider infinite dimensional Lie algebras of generalized Jacobi matrices gJ(k) and gl1(k), which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra gJ(k) and of the orthogonal and symplectic subalgebras. 1. Introduction. In this expository article, we consider a special type of general linear Lie algebras of infinite rank over a field k of characteristic zero, and compute their homology with trivial coefficients. Our interest in this topic came from an old short note (2 pages long) of Boris Feigin and Boris Tsygan (1983, [FT]). They stated some results on the homology with trivial coefficients of the Lie algebra of generalized Jacobi matrices over a field of characteristic 0. It was a real effort to piece out the precise statements in that densely encoded note, not to speak about the few line proofs. In the process of understanding the statements and proofs, we got involved in the topic, found different generalizations and figured out correct approaches. We should also mention that their old results generated much interest during these 36 years. In the meantime, two big problems have beed solved, which helped 2010 Mathematics Subject Classification: Primary 17B65, 16S35; Secondary 16E40. Key words and phrases: Infinite dimensional Lie algebras, Lie algebra Homology, Cyclic and Hochschild Homology, smash product, spectral sequence. The paper is in final form and no version of it will be published elsewhere. [1] 2 A. FIALOWSKI AND K. IOHARA us to understand the statements and work out the right proofs. One is the Loday-Quillen- Tsygan theorem (see (3.1)) and the other is the generalization of the Hochschild-Serre type spectral sequence by Stefan (see (3.4)), without which the statements of the old note could not be justified. Consider the Lie algebra gJ(k) of generalized Jacobi matrices, namely, infinite size matrices M = (mi;j) indexed over Z such that mi;j = 0 if ji − jj > N for some N de- pending on the matrix M. The original Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix (a band matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.) It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel mea- sure. This operator is named after Carl Gustav Jacob Jacobi, who introduced in 1848 tridiagonal matrices and proved the following Theorem: every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix (see [K] and e.g. [Sz]). Since then, Jacobi matrices play an important role in different branches of mathematics, like in topology (Bott's periodicity theorem on homotopy groups), stable homotopy theory, algebraic geometry and C∗-algebras (see [Ka1]). They are also used to show some inter- esting properties in K-theory. For instance, Karoubi used them to prove the conjecture of Atiyah-Singer about the classifying space, see [Ka2]. The Lie algebra gl1(k) of finitely supported infinite size matrices indexed over Z can be naturally viewed as a subalgebra of gJ(k). The Lie algebra gJ(k) has typical infinite dimensional nature. For example, the two matrices X X P := iEi−1;i;Q := Ei+1;i; i2Z i2Z where Ei;j is the matrix unit with 1 on the (i; j)-entry, belong to gJ(k) but not to gl1(k). They satisfy 1. tQ · Q = I, 2. PQ − QP = I, P where I denotes the identity matrix Ei;i 2 gJ(k). This matrix I does not even i2Z d belong to gl (k) ! (Off course, the matrices P and Q are matrix representations of 1 dt ±1 i and t· on the vector space C[t ] with respect to the basis ft gi2Z.) Such algebras show up in many areas of mathematics and physics. For instance, they are used to describe the solitons of the Kadomtsev-Petviashvili (KP) type hierarchies [Sa] where such integrable systems are interpreted as a dynamical system on the so-called Sato Grassmannian. On the other hand, their basic algebraic properties and invariants are not well understood. In our work we present results on their homology with trivial coefficients, For this, we need to use several different (co)homology concepts. Some results were obtained by Feigin and Tsygan [FT] in 1983, but in their short note the statements and proofs are not precise. In our work, we were able to get straightforward statements and proofs and we also could generalize the results to the coefficients over an associative unital k-algebra. GENERALIZED JACOBI MATRICES 3 The structure of the paper is as follows. In Section 2 we introduce some important classes of Lie algebras of general Jacobi matrices and recall their universal central ex- tension. We also give some examples of its subalgebras. Section 3 is devoted to their (co)homology. First we recall the main definitions: Lie algebra homology, Hochschild ho- mology, cyclic homology and (skew)dihedral homology. Then we compute homology with trivial coefficients for the introduced Lie algebras. In this section, we also present precise proofs of some results in [FT] and give possible generalizations. In Section 4 we introduce two important subalgebras, the orthogonal and symplectic subalgebras. To compute their (co)homology, we need to introduce additional computational methods to the previous one. In Section 5 we discuss a more general case, where instead of the field k we have an associative unital k-algebra, and generalize our (co)homology results for such algebras. Finally, we introduce a rank functional on a Lie subalgebra gJ1(k) ⊆ gJ(k) and describe its image. After describing its cohomology ring, we also raise some open questions. 2. Lie algebras of generalized Jacobi matrices. 2.1. Lie algebras gl1(k) and gJ(k). The first such Lie algebra one may have in mind is the one defined as an inductive limit: let I be a countable set and I1 ( I2 ( ··· ( In ( ··· , S I = n In an increasing series where each In is a finite subset. The Lie algebra glI (k) is defined by the inductive limit of glIm (k) ,! glIn (k) for m < n, namely, each element of glI (k) is a finitely supported matrix of infinite size indexed over I, i.e., X = (xi;j)i;j2I 2 such that ]f(i; j) 2 I j xi;j 6= 0 g < 1. In particular, for I = Z, we may denote by gl1(k). Another one we may also encounter is the Lie algebra of generalized Jacobi matrices defined as follows. A generalized Jacobi matrix is a matrix M = (mi;j) indexed over Z such that there exists a positive integer NM satisfying mi;j = 0 8 i; j such that ji − jj > NM : The set of such matrices has a structure of associative algebra over k denoted by J(k). We shall denote it by gJ(k) whenever we regard it as Lie algebra. An original Jacobi matrix is a finite size matrix M = (mi;j) such that mi;j = 0 for any i; j with ji − jj > 1. The Lie algebra gl1(k) can be naturally viewed as a subalgebra of gJ(k). 2.2. Universal central extension of the Lie algebra gJ(k). In the course of study- ing soliton theory, a non-trivial central extension of the Lie algebra gJ(k) was discovered (see, e.g., [JM] and [DJM]) and it can be described as follows. P Let J = i≥0 Ei;i 2 gJ(k) be a matrix and let Φ : J(k) ! J(k) be the k-linear map defined by X 7! JXJ. It can be checked that, for any X; Y 2 J(k), the element [Φ(X); Φ(Y )] − Φ([X; Y ]) is an element of gl1(k), i.e., only finitely many matrix entires can be non-zero. Hence, one can define the k-bilinear map Ψ : J(k) × J(k) ! k by Ψ(X; Y ) = tr([Φ(X); Φ(Y )] − Φ([X; Y ])); P where tr : gl (k) ! k ; X = (xi;j) 7! xi;i is the trace of finitely supported matrices. 1 i2Z It turned out that this Ψ is a 2-cocycle, called Japanese cocycle, i.e., 1.Ψ( Y; X) = −Ψ(X; Y ), 4 A. FIALOWSKI AND K. IOHARA 2. Ψ([X; Y ];Z) + Ψ([Y; Z];X) + Ψ([Z; X];Y ) = 0, for any X; Y; Z 2 J(k). Let gfJ(k) := gJ(k) ⊕ k · 1 be the Lie algebra whose Lie bracket [·; ·]0 is given by [X; 1]0 = 0; [X; Y ]0 = [X; Y ] + Ψ(X; Y )1 X; Y 2 gJ(k): As the Lie algebra gJ(k) is perfect, i.e., [gJ(k); gJ(k)] = gJ(k), the Lie algebra gJ(k) admits the universal central extension. It was B. L. Feigin and B. L. Tsygan [FT] in 1983 who proved that this central extension α : gfJ(k) ! gJ(k) is universal, namely, gfJ(k) is perfect and for any central extension β : a ! gJ(k), there exists a morphism of Lie algebras γ : gfJ(k) ! a such that the next diagram commutes: α gfJ(k) / gJ(k) A γ β a Remark 2.1.
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